
QA 

309 




LIBRARY OF CONGRESS, 1 



Chap. .(XA-^D.S-. 
Shelf ._.,£ .IX.. 



% UNITED STATES OF AMERICA. 












A MM of Integrating the Sanare Roots of Qoaflratics. 



By HENRY T. EDDY, C. E„ Ph. D., 



Assistant Professor of Mathematics in Cornell University , Ithaca-, N. Y. 



From the Proceedings of the University Convocation, i^eld at Albany, N. Y., 
I ^\ August 6th, 7th and 8th, 1872. 




b 






Kj—O^- 




\) 



- ft 






Integrating Square Roots of Quadratics. 257 



A METHOD OF INTEGRATING THE SQUARE ROOTS 

OF QUADRATICS. 

By HENRY T. EDDY, C.E., Ph.D., 

ASSISTANT PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY, ITHACA, N. T. 

CONTENTS. 
PART I. Upon J sin p cos q (a) 

PART II. Upon f(a + bx + cx*)*x m dx (h) 

PART I. 

To integrate the general form, / sin p <9 cos q 0<#0 (a). 

in all cases where p and q are integers, positive or negative. 

Apply to (a), the formula for integrating by parts, viz. : / udv = 

vn — / vdu, by making u = sin p-1 cos q-1 and dv = $mdcosd; 

sin 2 cos 2 sin 2 .... , , /iN 

.;. t> = , or = » — . It v == , we obtain formula (1) 

cos 2 
of the succeeding summary ; but if v = ^-— , we then obtain 

formula (2). 

Again, since 1 = sin 2 -f- cos 2 0, / sin p cos q 0dd, 

=J sin p + 2 oos^ddO + J sin p cos q + *ddd . . . . (c) 
—J sin p 0cos q ~ 2 0d0 — ysin p + 2 0cos q -W0 ...(d) 

.— y sin p - 2 0cosW0 — / > sin p - 2 0cos ,J + W0 ...(e) 

1st. Substitute in place of the last term of (c) its value as given by 
(1), from which process we shall obtain formula (3) of the summary. 

2d. Substitute in place of the last term but one of (c), its value as 
given by (2), from which we shall obtain formula (4). 

3d. Substitute in place of the last term of (d) its value as given by 
(2), from which we shall obtain formula (5). 

4th. Substitute in place of the last term of (e) its value as given by 
,(1), from which we shall obtain formula (6). 



258 University Convocation. 

SUMMARY. 
J sin p #cosW0 [a) 

— ——= sin p - *0 cos* ~*B + ^-^- / W ' 2 <9 cos* " W0 3 

= -smr-^cos*-^ +-^- z1 t fsmv-zdcos^Wde . . . . g 

— sin p - *0 cos* + *0 + i-? + g ~ - ' / W - 2 # cos*0rf0 . . . . (3) 

p + 1 P + 1 t/ 

= ^— -sin^^cos^e + p ^ - ~ /Wtfcos*~«0d0 . \ . (4) 

g + 1 g + 1 J w 

= sin p - 1 0cos <1 - 1 + ^^- I'&mvdcosi-zedd (5) 

p -r q p-t-JlJ 

= — sin p - 1 0cos ,1 + 1 + ^— ^ f'mi*-2ecQ&dd6 (6) 

p-hq p + qJ v 

Iii attempting to integrate the expression, / sin p # costfdd . . . 

(when p and q are integers, either positive or negative*), there can ouiv 
nine cases arise. These we shall now discuss seriatim, and, by sho 
them each to be integrable, prove the original expression integrable. 

1. If p and q are both positive integers, one odd and the other even. 
by repeated applications of (5) and (d). [a) will depend either upon 

/ miOdd = — cos0, 

or upon / cosddd = sin#. 

2. If p and q are both positive and even, by (5) and (6) finally we 
have only to integrate 

fdB = 6. 

3. If p and q are both positive and odd. by (5) and (6), (a) finally 

depends upon 

/» . . „ ln sin 2 cos 2 *? 
/ sm0 cosddd = — — , or = ^— . 

4. If p and q are both negative integers, one odd and the other 
even, by (3) and (4), (a) will depend either upon 

r dd rmkOffl Ir r wafidfl r mWi l. rl— cgs^h 

»/ smfl ~t/ s in 2 2 ~"2U' 1— cos# + ^ l + cos0J~~ 2 ° ge Ll-f cos0. 

ft 
= log e [cosec0 — eotang0] = log. tang - , 



Integrating Square Roots of Quadratics. 259 

or upon 

/' dd P cosddd 1 r P cosddd P cosddd [ _ 1 r l + sinfl 

«/ cos# ""«/ cos 2 ~2Le/ T+sin0 + e/ l — smflJ - ^ ° ge Ll — sinfl. 

((9 — • \ 

5. If j» and q are both negative and even, by (3) and (4), (a) will 
finally depend either upon 

/ sS> = - cotang "' orupon JJk = tmse - 



6. If p and g are both negative and odd, by (3) and (4), (a) will 
depend upon 

P dd _ P (sm 2 d + cos 2 #) _ Cf^LdB P— -dd 
J sinflcosfl """ «/ sin^cosft ~V cos# «/ sin0 

= loge sin# — log e cosfl — log e tang<9 = — Iog e cotangfl. 



7. If jo and </ are one odd and the other even, one positive and the 
other negative. 

When the odd number is numerically the larger, by (1) and (2), (a) 
will depend upon one of the four following forms (n being an integer) : 



Pwa** + *6dd, Pcos* n + 1 ddd, P-.-£L— 9 f 
J 'J J sm 2n + 1 # e/ 



dd 



sin 2n + 1 0' J cos 2n+1 ' 

which forms have been already integrated. 

When the even number is numerically the larger, we have similarly, 
by (1) and (2), four forms: 

Pm$»ddd , . , _ ... , _ P dd (6 n\ 

J ^oW~> WhlCh ' by (6)? dep6ndS ° n J ^0 = h & ta M2 + l)> 

Pcos^edd a i( u p dd ■ e 

fiSSi' " " (4) ' " -'/*«==-«* 

8. If ^ and q are both even, one positive and the other negative, by 
(1) and (2) we have the four following forms: 



fmP*dd6, f-~- a fcos 2n 6dd, I 



"' dd 



sin 2n t/ J */ cos 2n ' 

which forms have already been integrated, 



9. If p and 9 are both odd, one positive and the other negatire, bj 



/ 



"' :._-.:: ~ 



sin# : '-ZL ~ 



5 . * * y 



L0#5 






- - . . j: - z 
I '" '" " « - C / 

« el**- -7 ' '- " ■/ ' 

When. however, (p -f 0;) = 0, tike above are failing cases of (5) and 
(6), but not of (1) and (2}. 

SUMMARY. 

VTr :-.i. '::;:: :t. :j :it - :: : . ::_ -.:: 1 :■: : . ;:i:':i::~^. :■:.-. ^ 

yVjiri.f =--=-. : 

/ losrir = £L-f 5 

-^=-- : - :: ^ * 

-^^ = 106,^ 

1 ...:"■. ^ = 3*g. rangS = - ^g* cotangf (13) 

fae = $ (u) 



— x — 



a. /"_*_ = 



. .'_i. i, - 



IHTEQUATING Si^UAtiX HOOTS OJP Q VADKATW&. 2(J1 

r sinOcld _ 1 

J cos 2 cos# * 

/"GosddO _ 1 

J sin 2 ~~ sin0 * 



5. fmPBdd = hd — sintf cos0). 

6. fco&dde = i (0 + sin0cos0). 

7 ' / art - — tal1 ^ — ft 

«/ cos 2 6 



*/* 



cos 2 6dd 
. __ = — cotangfl — 0. 
sin 2 © & 



/* 1 1 

10. / sin0cos0tf0= o sin2 ^ or = — «cosW. 

•^ <© /& 

It is possible to apply each of the formulae from (1) to (6), inclusive, 
to that part of itself which is still under the sign of integration, or to 
the corresponding part of either of the others. This operation can be re- 
peated at will, and we thus obtain formulae which are some of them 
useful, as they give at once the integral of expressions which it would 
otherwise require several processes to effect. 

By repeatedly applying each of the formulae (1) to (6) to itself, as 
suggested, we have the general forms of such expansions in the six 
following formulae^ which we arrive at after effecting n integrations 
(n being an integer greater than unity). These formulae may be estab- 
lished by mathematical induction. 

y sin p co&ddO, 
by (1) : 

= --Ur waF + ^6 cos*-^ + ^~^ • _-!_— sin p +8 cos*-*0 + 
P + 1 (P+ 1 ) (jP + 3) 

0>+l) 0> + 3) tP + 5) T T 

(q— l)( g — 3) [q + l—2(n—l)] sinP- 1 + 2n 0cos^ + 1 - 2n 

0> + i)(p + 3) [>-l + 2(rc-l)] " [^ + 1 + 2(^-1)] + 

(P + l)(p + 3) (^-1 + 2^)^ v J 



262 University Convocation. 

by (2) : 

= — r sin p -^ cos*- 1 ^ — ( /~J • - — sm*-sd cos* + 3 — 

q + 1 (A+ 1 ) (5't3) 

(^.lrf._J__ sill ,-5, C0 ^- 5 ,_ I 

(£+1) t? + 3) fe+5) 

(j? — 1) (p — 3) [p + 1 — 2 ( /v — 1) ] sinP- 1 - 2 ^ cos 9-1 ^ 

(<7 + l)(? + 3) [g _ l + 2 (n - 1) ] •-j^+i + 2(»^l)J"~ 

^-jl^ ^!~^ /W"W«-iii . . (16) 

(g + 1) (£ + 3) (2—l+2»)« / v 

by (3): 

= — ^ ?in- D - *0 cos* + *0 + ^ + g "t ? ^ • , 1 ox sin* - 3 cos* ^6 + 

l? + i (i? + 1) (.p+3) 

(^ + g + 2)> + g + 4) _j_ ^ _ 

(i^ + l) CP + 3) tp+5) + 

(jg + g + 2) (p + gr + 4) l> + g + 2 (n — 1)] sin^ 1 -^ cos^fl 

CP+1) (jP+3) 0-l+2(»-l)]"l>+l+2(»-l)] + 

(P+l)(jP+3) |>+l+2(»— 1)]«/ ^" 

by (4) : 

= — -smP-^cos^^- ^" 1 "^^ 2 ^ . * o , sm^flcos^ 3 - 

g + 1 . (2 + 1) (g+3) 

(FW»)>+g+4) a _l ^ + + ...... 

(0 + 1) (?^3j (gr + 5) 



Cp + g + 2)(j> + g + 4) [yj + g + 2 (;*-!)] gin^-^cos^^-»0 

( g -l)( g + 3) [gr_l + 2(«-l)] '[? + l + 2(»_l)] + 

te+ 1 )te+ 3 > b+l+8(»— l)]v v ' 

by (5): 

= sin p " 1 #cos <1 - 1 f9 + ; g ~ ; - 7 -sin^flcos*"^ + 

l> + (jP + ?) (^ + ^-2) 

(£z^.Jt^._J_ a i n ^^- 50+ + 

( g _l)( g _ 3) [g + 1— 2(n— 1)] sin^^cos^ 1 -^ 

te + gOCp + 2-2) [ p -L. q -2-2(?i-D] ' [p-g-2 (, ? -l)] + 

7 ( f- 1)(g ~f (^ + 1 ^ faa*Ooo#-*H>dB (19) 



Integrating Square Roots of Quadratics. 263 

by (6): 

sin p ~ J cos q + 1 6— Y~~ \ • 5 ^- sin p - 3 cos q+1 — 



^ + g (jp + gr) (jp + gf— 2) 

(p-1) (p-S) 1 



sin p - 5 0cos q + 1 — — 

anP+i-fctf cos q+1 

+ 



(i>+2) (p+q-%) (p+q—±) 

(^_l)( 7 j_3) [^4-1—2(^—1)] s i n p+i-2M C0S g+i(9 

(p + q)(p + q-%h. • .[p + q + Z-2(n-l)]' [p + q-2(n-l)] 

( pl) (g 8 ) . [^ + 1-^ 1 fte-t-BvXW (20) 

If formulae (1) to (6), inclusive, may be called primaries, we shall 
now give six secondary formulae, (21) to (26), inclusive, which can be 
obtained by one operation of one of the primaries upon one of the 
others. These six (viz. : (21) to (26) ), together with the six contained 
in the general formulae (15) to (20), inclusive, are all the secondaries 
obtainable, and they may serve as an example of the tertiaries, &c, 
possible : 

y*sin p 0cosW0, 

by applying (1) to (3) • 

sin p + W cos q ~ !0 + ? ^~ ' _. sin p + 8 cos q ~ x -f 



by applying (2) to (4), 

sin^flcos 15 -^ — - — L^zl) g m P-i(9 C0S q+85 + 



q,+ l (q + l){q + 3) 
^/~ 1 l4 J? / +g ^ 2; Ain p -26>cos q + W0 (22) 

by applying (1) to (5) : 

sin p + 1 0cos q ~ 1 + t — w ^ sin p + 1 0cos q - s + 



p + q (p+qKp+V 

(g— l)(g— 3) / , s in v + 2o co &-4edd (23) 

(p+q){p+ 1 ) J } 

by applying (2) to (6): 

sin p ~ *0 cos q + x — - — '^~ ' sin p - 8 cos q + *0 + 



p + q (p+q)(q+i) 

(p-l)( p -3) /* inP -40 co g» + 80^ (24) 

(p+q){q+i) J 



264 University Convocation. 

by applying (3) to (4) : 

sin p - l d cos q - 3 ■ — sin p + 3 # cos* * *fl -f 



feHH^t^ y-^^^^ 

by applying (5) to (6): 

%mr^d cos*-^ r~ — sin* - 1 6 coeP"0 + 



{p+q){p+q— 2) (i>-f?)(i>4-?-2) 

, ^"l^?" 1 ^ fsiD*-*dcos*-*ede (26j 



Integrating Square Roots of Quadratics, 265 



PART II. 



To integrate the general form, u = / (a + bx + ex 2 ) 2 x™dx . . (b) 

in all cases when m and n are integers, positive or negative, provided 
the constants a, b, and c are such as to render the integral real. 
By change of form, we have, if 

u — J (a -f bx + cx 2 )2x m dx (J) 

u=V?j\{x + Tc ) j^^dx (h) 

u =v^f[(^+ij-^p^ . . .(*) 

We shall now point out briefly what transformations will cause 
these forms to depend for their integration upon that of (a). 

1st. Formula (/) may be integrated when m is positive and 

7)2 A-clc b 

r 2 = — -r~2 — > ; for, let x -+- — = r sin# ; . • . dx = r cosddd, and 

rb 2 — 4:ac I b 2 \ 2 ~X?- 
— j~2 — ( x + — ) p = r n cos n 0. Substituting, we have, 

u = r n+m+1 V^^f [sin0 — /-Teos" + W0 . . . (27) 

which is real when c is negative, and is integrated by expanding, mul- 
tiplying, and thus obtaining several terms of the form of (a). 



266 University Convocation. 

» 
This transformation might have been effected with equal ease by 

cutting x + — = r cos#, etc., etc. 

2d. Formula (g) may be integrated when m is negative, and 

J2 _ 4_ ac ft fa 

H = — — — > ; for, let or 1 + 7- = r sin0 ; . •. - — r cosddO. 

4or Act x z 

and I — j~2 ( x' 1 + — j J2 = r n cos n 0. Substituting, we have, 

u = — r-™' 1 V — a n J I sin0 — ^— concede . . (28) 



which is real when a is negative, and it is integrable whenever we can 
make — (m -f n + 2) > 0, in which m is negative. This we can 
always effect by making the exponent of the quadratic any minus 
number desirable, as follows : 
Multiply and divide (g) by 



. 4a 2 V + 2c/J 



in which — (m — s -f 2) > 0, and expand the numerator to the power 

S I fh 

indicated by the exponent — - — (which exponent must be an integer 

when 5 and n are odd). Thus we obtain several terms, each of which 
can be integrated by (28). 

3d. Formula (h) may be integrated when m is positive and 

r 2 = — 7-= — '- > ; for, let x + tt~ = r sec<9 > .'. dx= r— - , and 

4c 2 2c cos^0 

r/ h \ 2 & 4#c~i- 

i j a; + 75- ) jl — 2 = r " * an g n ^- Substituting, we have, 

w = r n+m+ V c n / sec<9_ — - — 2.-- . . . (29) 

«/ L 2crJ cos0 v 7 

which is real when c is positive, and depends upon (a) in the same 
manner as (27) and (28). 

4th. Formula (Jc) may be integrated when m is negative, and 

i 2 — kac -~ -.■,»,/*.# \ n dx r sinddd 

; ' 2= -^- >0: for ' let ( 3r '+W = '- sec9; •'• -? = T* 

K$ \ 2 $2 4#c"l- 
x~ x + — ) j-g — 2 = r B tang n #. Substituting, we have, 

. 1 /— /T o * l- (m+n+2) tang n - ) - 1 (9^ 

which is real when a is positive, and it is integrable when we render 
(as previously) — (m + w + 2) > 0. 



Integrating Square Roots of Quadratics. 26? 

5th. Formula (h) may be integrated when m is positive and 

b 2 — 4:ac b n _ rdB 

r 2 == j-= — > ; for, let x + — = r tangfl ; . •. ofe = — -. Sub- 

stitute, and we have, 

— V7/ N 4J^ . . . ( 3X) 



u = r 



6th. Formula (&) may be integrated when m is negative and 

5 2 — 4ac „ . 5 . da; rd0 

^=--4^->0; for let xi + - = r imgf>; .: -^ = 535- 

Substitute, and we have, 



w 



= _, m V«»ytang^_J — j- . . (32) 

SUMMARY. 
u = J {a + bx + cx 2 )2x m dx (5) 



If x + — = r sinfl, 







w 


^inJ-n-j-l 


V- 


C n 


If 


ar 


^2a 
u 


= r sin (9, 
r 








m+l ( 


/ 


If 


a; 


b 


r seed. 







~sin0 — —-1 cos n+ W0 .... (27) 



p-(in-f-n+2) 



2ar_ 



cos a+1 ddd . . (28) 



V e/ Lcos# ~ 2ir] Qos^d ' ' ' * ^ y > 



7 

If a;" 1 + — = r seed, 
Za 



u 



yV /»r 1 & -i-<-+ B +*> gin B+1 gdg 
— — ^m+i,/ Lcos0 ~~ 2arJ cos n+2 <9 ' '■ *'* ' 



If a; -f — = tang0, 
Zc 



J Lcostf 2crJ cos n+2 l ' 



If x~ x + — : = r tangtf. 



b_ 
2a 

V~aF nsmd b -7-( m + n + 2 > dd 



u 



— — rm +iJ [ cosd ~ 2^ I C os n+2 ' ' ' ' \> 



268 University Convocation. 

EXAMPLES. 

1. J (a + bx + cz*)2(h + kxydx. 

In this let h +kx = x l ; then the integral will be of the form {b), 

if p and q are integers. 

/p. ± 

[a + fo;)2(A + kx)2afdx 

[(a + to) (h + fe»)p(A + fcuJ'Ts-tfa;, when ^-^ > ; 
or, 

—J [(« + ox) (h + ^)2(fl + bxf^xTdx, when ^^ > 0. 

Expand the binomial to the power + * -. Since that is a posi- 
tive integer, multiply and integrate the several terms each of the 
form (b). 

(a -f cx 2 )2(h + ex^afdx may, in many cases, be transformed 
into (a) either by making (a -f- c.t 2 ) = r 2 tang 2 and r 2 = — =- (aA — ce) 

/i 

^ 

or (az~ 2 -f c) = r 2 tang 2 # and r 2 = -(ah — ce). A full discussion of 

this form might be made, similar in nature to the discussion of 
form (b). 

(1 + cx 2 )x^-' 2 dx p (x~ 2 + c)dx 



4. 






(l—cx 2 Y(l + c^ + c 2 ^ 4 ) 8 t7 (ar* — cxc)« [(a;" 1 — c2-) 2 + a + 2c] P * 
In this let (a: -1 + ox) = r tangfl and r 2 = a + 2c. 
(J -j- ere 2 -|- bc 2 x i )x p ~ l dx 



'■ / 



(1 — c 2 * 4 ) (1 + arc 2 + c 2 ^) 9 
_2fc + e /» (l + cay-^a 2fa— e n (l—ca?)a? - l dx 

4c ^ (l—cx*)(l + ax 2 + c 2 x i )% 4C ^ (l+^)(l + aa?+c^ 

which is a case of Example 4. 

It may be stated, in conclusion, that the method herein briefly 
sketched, by which integrals containing some power of the square root 
of a quadratic are transformed and made to depend on form (a), has 
been found by the author, in practice, to be practicable, expeditious, 
and useful, especially when a proper transformation of limits is effected 
at the same time as the first-mentioned transformation. 



Integrating Square Roots of Quadratics. 269 

NOTE A. 

The following tables of relations between circular functions will 
facilitate the necessary transformations. 

TABLE I. 
Relations between the Direct Circular Functions. 



sin0 = 


= Vi 




tangfl 


1 


Vs 




-1 1 1 




,ec 2 0- 


— COS 2 = 


Vl+tang 2 


Vl + cot 2 




sec0 


cosec0 






cos0 = 


1 


cote 




1 

sec0 






Vcosec 2 0— 


■1 


Vl — sin 2 0= 


Vl+tang 2 


Vl + cot 2 


■ cosec0 




sin0 


Vl 




= tangtf = 


1 

cot0 


= V* 




1 




— COS 2 

cos0 


sec 2 0- 


-1- X 


71— sin 2 


I Vcosec 2 0— 


•1 






cos0 


r 

tang# 


= cot0 = 




1 






VI— sin 2 




1 


sin0 


Vl 


— cos 2 


V* 


sec 2 0- 


-1 | 


1 




1 

cos0 








sec0 


cosec0 




Vl + cot 2 
cote 


= Vl + tang 2 6 = 


Vl— sin 2 


Vcosec 2 0— 


■1 


1 

sin<9 




1 








sec0 


1 

— cosec0 




Vl+tang 2 
tang0 


= Vl + cot 2 = 


Vl 


— cos 2 


Vsec 2 0— 


-1 



TABLE II. 
Relations between the Inverse Circular Functions. 



sin- 1 ^ = cos~ x Vl — x 2 = tang- 1 = cot 



Vl— £ 



Vl— a 2 



:sec 



-i 1 



vr 



t i 

cosec -1 - 



Vl— a 2 



sin - Vl — x*= cos -I £c =tang -1 =cot -1 



Vl— a; 2 



L l I , 1 

sec -1 - =cosec _1 

a? ! 



Vl— j 



sin~~ l — : =cos~ 1 = tang _1 a; 

Vl+x 2 Vl+as 



= cot -1 - =sec- 1 V4-la; 2 = cosec- 1 



Vi+; 



1 ' x I 1 

sin— 1 ^cos -1 - = tang- 1 - 

Vl+Z 2 . Vl+CC 



cot -1 a; 



Vl-fa" I , 

sec -1 = cosec -1 V 1 -f x~ 



Vx 3 -1 I 1 I { l - r — 

sin -1 = cos- 1 - =tang — War — 1: 

® i x \ 



cot - 



Vr-1 



= sec"" 1 ^ =cosec" 1 



Vaj a — 1 



sin -1 - =cos 

x i » 



Vcc 2 -1 

— =tang _1 



Vz 2 -i 



=cot~ 1 Va; 2 — 1 = 



sec - 



cosec a: 



Vz 2 -i 



270 University Convocation. 

NOTE B. 

The symmetrical manner in which formulae (1) to (6^, inclusive, are 
obtained may be further shown by the following process of obtaining 
the ordinary reduction formulas for the form, 

J X v x m ~ x dx - . . (b 1 ) 

in which X = a ; + bx°. Integrating by parts, the parts being indi- 
cated by the period, 

fx*. x™~Hx = ?^ - — V - fx*- x x M - l dz (1) 

or, 

x m ~" . Xx^dx = - , ^ - , / X^'x^-'dx . . (2) 

bn(p + l) bn{p-hl)J v ' 

It is to be noticed that (1) and (2) are the only integrations by 
parts of (J) 1 ) which are binomials. Again, we have identically by 
separation, 

J Xfx^dx — a J X^-hp-Ux -f- bj X p -V ,+a -^a; . . (s 1 ) 
JX p+1 x m - 1 dx=zaJ'XPx m - 1 dx + bJ'xvx m+n - 1 dx . . . (s) 

yp^" 1 - 1 ^ = afXFtf^-Wx + bfx*x™-Wx . . . (s,) 

It is to be noticed that (s 1 ), (5), and (s t ) are the only separations of 
(b 1 ) which are binomials. 

If now the last term of (s 1 ) be integrated by parts in the same 
manner as (2), we shall obtain (5) of the following summary, after 
solving for (b 1 ). 

If the last term of (s) be integrated by parts similarly to (2), we 
shall obtain (4). • 

If the first term of (s) be integrated by parts . similarly to (1), we 
shall obtain (3). 

If the first term of (si) be integrated by parts similarly to (1), we 
shall obtain (6). 

It is to be noticed that (3), (4), (5), and (6) are the only integra- 
tions by parts of (s 1 ), (s), and (sj which are binomials; 



Integrating Square Roots of Quadratics. 27 J 

SUMMARY. 

/x p r-% (b 1 ) 

XV hip /\ rn < - , „ < 7 

= 7-7— TTx - x-7 — r-Tx / XP +1 ^ m - n -i^; ... (2) 

X p+ V J(m + »+»») /»_ 

=± — — — --^- / XV^dx ... (3) 

— X p+ V' m+n + np f* ,. 

an(p + 1) an (p -j- 1) t/ w 

X p z m <m« /*„ i 
= 1 =£- / X v -hf a ~ 1 dx ..... (5) 

X p+i^ m -n a{m—n) /•_ 

= Tr i ^ — TT— i T / XV-"" 1 ^ ... (6) 

6(m + rap) 6(m + rap) t/ v ' 



IB 



